The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was \(2.5\) stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF). The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
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Add up the ring widths to get every boundary radius, then use pi(R_out^2 - R_in^2) for each ring's area before forming the ratio.
Step 1: Use the sum-times-difference shortcut for ring area. A ring between radius $R_{in}$ and $R_{out}$ has area $\pi(R_{out}^2-R_{in}^2)=\pi(R_{out}+R_{in})(R_{out}-R_{in})$. The second factor is just the given width of the ring, so this avoids squaring big decimals.
Step 2: List every boundary radius. Starting from the centre: $0,\ 2.5,\ 3.5,\ 5.5,\ 7.5,\ 10.5,\ 13.5$, built by adding the Inner Island radius and then each ring width in turn.
Step 3: Apply the shortcut to each ring. Water ring AB: $(3.5+2.5)(1)=6$. Land ring BC: $(5.5+3.5)(2)=18$. Water ring CD: $(7.5+5.5)(2)=26$. Land ring DE: $(10.5+7.5)(3)=54$. Water ring EF: $(13.5+10.5)(3)=72$. Each of these is the bracket value only; multiply by $\pi$ for the actual area.
Step 4: Add the Inner Island and total each side. Land (in units of $\pi$): $2.5^2+18+54=6.25+18+54=78.25$. Water (in units of $\pi$): $6+26+72=104$.
Step 5: Take the ratio. \[ \frac{78.25}{104}=0.7524\ldots \approx 0.75 \] \[ \boxed{0.75} \]
